Linear preservers of parallel matrix pairs with respect to the $k$-numerical radius

Let $1\le k<n$ be integers. Two $n\times n$ matrices $A$ and $B$ form a parallel pair with respect to the $k$-numerical radius $w_k$ if $w_k(A+\mu B)=w_k(A)+w_k(B)$ for some scalar $\mu$ with $\mathrm{\mid }\mu \mathrm{\mid }=1$; they form a TEA (triangle equality attaining) pair if the preceding equation holds for $\mu =1$. We classify linear bijections on ${\mathbb{M}}_n$ and on ${\mathbb{H}}_n$ which preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of $w_k$-isometries, except for some exceptional maps on ${\mathbb{H}}_n$ when $n=2k$.